### Problem Statement**2) Determine if the following series converges or diverges using the Integral Test. Show your work carefully and detailed.**\[ \sum_{n=1}^{\infty} \frac{e^{1/n}}{n^2} \]### Detailed Explanation#### Integral TestTo apply the Integral Test, we need to check if $ f(x) = \frac{e^{1/x}}{x^2} $ is continuous, positive, and decreasing for $ x \geq 1 $. **Step 1: Verify if $ f(x) $ is continuous, positive, and decreasing**- **Continuity**: $ e^{1/x} $ is continuous for all $ x \neq 0 $ and $ x^{-2} $ is continuous for $ x \neq 0 $. Hence, $ f(x) $ is continuous for $ x \geq 1 $.- **Positivity**: $ e^{1/x} > 0 $ and $ x^{-2} > 0 $ for $ x \geq 1 $, thus $ f(x) > 0 $ for $ x \geq 1 $.- **Decreasing**: To check if $ f(x) $ is decreasing, we can find the derivative $ f'(x) $ and see if it is less than or equal to 0 for $ x \geq 1 $.**Step 2: Find $ f'(x) $**Let $ f(x) = \frac{e^{1/x}}{x^2} $.Using the quotient rule for derivatives:\[ f'(x) = \frac{d}{dx} \left( \frac{e^{1/x}}{x^2} \right ) = \frac{(e^{1/x})' \cdot x^2 - e^{1/x} \cdot (x^2)'}{(x^2)^2} \]First, calculate $ (e^{1/x})' $:\[ (e^{1/x})' = e^{1/x} \cdot \left( \frac{d}{dx} \left( \frac{1}{x} \right) \right) = e^{1/x} \cdot \left( -\frac{1}{

Name: ### Problem Statement**2) Determine if the following series converges
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Description: ### Problem Statement**2) Determine if the following series converges or diverges using the Integral Test. Show your work carefully and detailed.**\[ \sum_{n=1}^{\infty} \frac{e^{1/n}}{n^2} \]### Detailed Explanation#### Integral TestTo apply the In

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Answered: Determine id Jhe follo wiyp series…

Math

Calculus

Determine id Jhe follo wiyp series Converpes or Yntearel Test. Show oliverpes using The your work carefule

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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